Chapter 3 1

Chapter 3

3.1 Simplification of Boolean Functions

• Easy method: Use pictorial form of a truth

table

– Karnaugh map

• Diagram of squares

• Each square has a minterm

• The map is a visual representation of a way a

function can be expressed

Chapter 3 2

• Any two adjacent squares differ by one

variable (primed in one unprimed in the other)

• The different variable is removed

• F = x’y + xy =

1) How do we represent this row? ->

x’

2) Another way is to use boolean

algebra x’(y+y’) = x’

Chapter 3 3

• 3-variable map

• Example: Simplify F = x’yz + x’yz’ + xy’z’ + xy’z

– Show the 1’s

– Enclose adjacent 1’s (power of 2)

Chapter 3 4

• Example 2

– Simplify F = x’yz + xy’z’ + xyz + xyz’

F =

Consider m0+m2+m4+m6

Q: What happens if we have all 8

squares combined?

A: Function is always equal to 1.

Chapter 3 5

• Example 3

– Simplify F = x’z + x’y + xy’z + yz

Chapter 3 6

• 4-variable map

• Example– Simplify F = A’B’C’ + B’CD’ + A’BCD’ + ABC’

Chapter 3 7

• Ensure all terms are covered

• Do not forget that it is necessary to minimize

the # of terms

• F =

• F =

• Essential Prime Implicants

Choosing adjacent squares

Σ(0,2,3,5,7,8,9,10,11,13,15)

Chapter 3 8

F =

• Product of sums

– Simplify F(A,B,C,D) = Σ(0,1,2,5,8,9,10)

– If we combine the 0’s: F’ =

Or (DeMorgan): F =

- Previous example: sum of products was used

Chapter 3 9

3.2 Don’t care conditions

• 4-bit binary code for decimal digits has 6

unused combinations

– There are applications in which we don’t care

what the value of the function is in these cases.

– These don’t care conditions may be used to

further simplify a boolean expression.

• Example: Simplify F(w,x,y,z) = Σ(1,3,7,11,15)

– Don’t care conditions: d(w,x,y,z) = Σ(0,2,5)

• To distinguish the don’t care conditions from the 0’s

and 1’s we use an X.

Chapter 3 10

• If we combine the m5 (=x) with the 1’s, then

• If we combine the 0’s, then

Chapter 3 11

3.3 NAND & NOR GATES

• NAND: Universal gate (any digital circuit can

be implemented using only NAND gates)

• We just have to show that AND, OR and NOT

can be implemented with NANDs

Chapter 3 12

• Implementation of F = AB + C with NAND gates

– Method 1:

– Method 2:

Chapter 3 13

• Logic operations with NOR Gates

Chapter 3 14

• Exclusive-OR

– x ⊕ y = x’y + xy’

Chapter 3 15

• Exclusive-OR

– x ⊕ y = x’y + xy’

Chapter 3 16

Example: Design a 3-bit even parity generator

Chapter 3 17

Example: Design a Parity Error Checker